Question

If $x=9$ is the chord of contact of the hyperbola $x^{2}-y^{2}=9$ , then the equation of the pair of tangents forming the chord of contact is $ax^{2}-by^{2}-18x+9=0$ . Find the value of $a+b$ .

Answer 1

The equation of chord of contact at point $\left(\right.h,k\left.\right)$ is
$xh-yk=9$
Comparing with $x=9$ , we get
$h=1,k=0$
Equation of pair of tangents at point $\left(\right.1,0\left.\right)$ is
$SS_{1}=T^{2}$
$\Rightarrow \left(x^{2} - y^{2} - 9\right)\left(1^{2} - 0^{2} - 9\right)=\left(x-9\right)^{2}$
$\Rightarrow -8\left(x^{2} - y^{2} - 9\right)=x^{2}-18x+81$
$\Rightarrow 9x^{2}-8y^{2}-18x+9=0$
$\Rightarrow a=9,b=8$
$\Rightarrow a+b=9+8=17$